E[t=0∑T∇θlogπθ(at∣st)b(st)]=0, ∇θJ(πθ)=E[∑t=0T∇θlogπθ(at∣st)∑t′=tT(γt′rt′−b(st))]=E[∑t=0T∇θlogπθ(at∣st)∑t′=tTγt′rt′]\begin{aligned} I think Sutton & Barto do a good job explaining the intuition behind this. ∇wV^(st,w)=st, and we update the parameters according to, w=w+(Gt−wTst)stw = w + \left(G_t - w^T s_t\right) s_t For example, assume we take a single beam. &= 0 \end{aligned}∇θJ(πθ)=E[t=0∑T∇θlogπθ(at∣st)t′=t∑T(γt′rt′−b(st))]=E[t=0∑T∇θlogπθ(at∣st)t′=t∑Tγt′rt′]. However, also note that by having more rollouts per iteration, we have many more interactions with the environment; and then we could conclude that more rollouts is not per se more efficient. The environment we focus on in this blog is the CartPole environment from OpenAIâs Gym toolkit, shown in the GIF below. ∇θJ(πθ)=E[∑t=0T∇θlogπθ(at∣st)∑t′=tTγt′rt′]\nabla_\theta J\left(\pi_\theta\right) = \mathbb{E} \left[\sum_{t = 0}^T \nabla_\theta \log \pi_\theta \left(a_t \vert s_t \right) \sum_{t' = t}^T \gamma^{t'} r_{t'}\right] The learned baseline apparently suffers less from the introduced stochasticity. REINFORCE has the nice property of being unbiased, due to the MC return, which provides the true return of a full trajectory. Self-critical sequence training for image captioning. Why? If we are learning a policy, why not learn a value function simultaneously? Vanilla Policy Gradient (VPG) expands upon the REINFORCE algorithm and improves some of its major issues. In the case of a stochastic environment, however, using a learned value function would probably be preferable. # - REINFORCE algorithm with baseline # - Policy/value function approximation # # ---# @author Yiren Lu # @email luyiren [at] seas [dot] upenn [dot] edu # # MIT License: import gym: import numpy as np: import random: import tensorflow as tf: import tensorflow. But wouldn’t subtracting a random number from the returns result in incorrect, biased data? Using samples from trajectories, generated according the current parameterized policy, we can estimate the true gradient. \end{aligned}E[∇θlogπθ(a0∣s0)b(s0)]=s∑μ(s)a∑πθ(a∣s)∇θlogπθ(a∣s)b(s)=s∑μ(s)a∑πθ(a∣s)πθ(a∣s)∇θπθ(a∣s)b(s)=s∑μ(s)b(s)a∑∇θπθ(a∣s)=s∑μ(s)b(s)∇θa∑πθ(a∣s)=s∑μ(s)b(s)∇θ1=s∑μ(s)b(s)(0)=0. However, the fact that we want to test the sampled baseline restricts our choice. Several such baselines were proposed, each with its own set of advantages and disadvantages. As maintainers of, and the first Ethereum client embracing Baseline, we are excited that the solutions delivered by Nethermind and Provide enable rapid adoption, allowing enterprises to reinforce … This is what is done in state-of-the-art policy gradient methods like A3C. … Simply sampling every K frames scales quadratically in number of expected steps over the trajectory length. Using the definition of expectation, we can rewrite the expectation term on the RHS as, E[∇θlogπθ(a0∣s0)b(s0)]=∑sμ(s)∑aπθ(a∣s)∇θlogπθ(a∣s)b(s)=∑sμ(s)∑aπθ(a∣s)∇θπθ(a∣s)πθ(a∣s)b(s)=∑sμ(s)b(s)∑a∇θπθ(a∣s)=∑sμ(s)b(s)∇θ∑aπθ(a∣s)=∑sμ(s)b(s)∇θ1=∑sμ(s)b(s)(0)=0\begin{aligned} Atari games and Box2D environments in OpenAI do not allow that. contrib. δ=Gt−V^(st,w), If we square this and calculate the gradient, we get, ∇w[12(Gt−V^(st,w))2]=−(Gt−V^(st,w))∇wV^(st,w)=−δ∇wV^(st,w)\begin{aligned} reinforce-with-baseline. A reward of +1 is provided for every time step that the pole remains upright. Then, ∇wV^(st,w)=st\nabla_w \hat{V} \left(s_t,w\right) = s_t Another limitation of using the sampled baseline is that you need to be able to make multiple instances of the environment at the same (internal) state and many OpenAI environments do not allow this. Some states will yield higher returns, and others will yield lower returns, and the value function is a good choice of a baseline because it adjusts accordingly based on the state. reinforce_with_baseline.py import gym: import tensorflow as tf: import numpy as np: import itertools: import tensorflow. frames before the terminating state T. Using these value estimates as baselines, the parameters of the model are updated as shown in the following equation. So far, we have tested our different baselines on a deterministic environment: if we do some action in some state, we always end up in the same next state. Switch branch/tag. The major issue with REINFORCE is that it has high variance. However, all these conclusions only hold for the deterministic case, which is often not the case. To tackle the problem of high variance in the vanilla REINFORCE algorithm, a baseline is subtracted from the obtained return while calculating the gradient. We have implemented the simplest case of learning a value function with weights w. A common way to do it is to use the observed return Gt as a âtargetâ of the learned value function. However, the unbiased estimate is to the detriment of the variance, which increases with the length of the trajectory. To reduce … Please correct me in the comments if you see any mistakes. After hyperparameter tuning, we evaluate how fast each method learns a good policy. Buy 4 REINFORCE Samples, Get a Baseline for Free! We use same seeds for each gridsearch to ensure fair comparison. For comparison, here are the results without subtracting the baseline: We can see that there is definitely an improvement in the variance when subtracting a baseline. If the current policy cannot reach the goal, the rollouts will also not reach the goal. We will choose it to be V^(st,w)\hat{V}\left(s_t,w\right)V^(st,w) which is the estimate of the value function at the current state. This can be a big advantage as we still have unbiased estimates although parts of the state space is not observable. ##Performance of Reinforce trained on CartPole ##Average Performance of Reinforce for multiple runs ##Comparison of subtracting a learned baseline from the return vs. using return whitening … The algorithm involved generating a complete episode and using the return (sum of rewards) obtained in calculating the gradient. Actor Critic Algorithm (Detailed explanation can be found in Introduction to Actor Critic article) Actor Critic algorithm uses TD in order to compute value function used as a critic. We would like to have tested on more environments. High variance gradients leads to unstable learning updates, slow convergence and thus slow learning of the optimal policy. Sensibly, the more beams we take, the less noisy the estimate and quicker we learn the optimal policy. The issue of the learned value function is that it is following a moving target, meaning that as soon as we change the policy the slightest, the value function is outdated, and hence, biased. The variance of this set of numbers is about 50,833. We focus on the speed of learning not only in terms of number of iterations taken for successful learning but also the number of interactions done with the environment to account for the hidden cost in obtaining the baseline. The following methods show two ways to estimate this expected return of the state under the current policy. This is also applied on all other plots of this blog. New campaign to reinforce hygiene practices in dorms Programme aims to keep at bay fresh mass virus outbreaks among migrant workers. Note that the plot shows the moving average (width 25). I do not think this is mandatory though. Shop leggings, sports bras, shorts, gym tops and more. This is why we were unfortunately only able to test our methods on the CartPole environment. Shop online today! This is called whitening. This indicates that both methods provide a proper baseline for stable learning. Developing the REINFORCE algorithm with baseline. This is a pretty significant difference, and this idea can be applied to our policy gradient algorithms to help reduce the variance by subtracting some baseline value from the returns. Contrast this to vanilla policy gradient or Q-learning algorithms that continuously increment the Q-value, which leads to situations where a minor incremental update … What is interesting to note is that the mean is sometimes lower than the 25th percentile. But in terms of which training curve is actually better, I am not too sure. Shop Baseline women's gym and activewear clothing, exclusively online. To find out when the stochasticity makes a difference, we test choosing random actions with 10%, 20% and 40% chance. However, we can also increase the number of rollouts to reduce the noise. It can be shown that introduction of the baseline still leads to an unbiased estimate (see for example this blog). Therefore, we expect that the performance gets worse when we increase the stochasticity. \nabla_w \left[ \frac{1}{2} \left(G_t - \hat{V} \left(s_t,w\right) \right)^2\right] &= -\left(G_t - \hat{V} \left(s_t,w\right) \right) \nabla_w \hat{V} \left(s_t,w\right) \\ … Stochastic Beams and Where to Find Them: The Gumbel-Top-k Trick for Sampling Sequences Without Replacement. Policy Gradient Theorem 1. We use ELU activation and layer normalization between the hidden layers. Nevertheless, there is a subtle difference between the two methods when the optimum has been reached (i.e. The REINFORCE algorithm takes the Monte Carlo approach to estimate the above gradient elegantly. Hyperparameter tuning leads to an optimal learning rates of Î±=2e-4 and Î²=2e-5 . Mark Saad in Reinforcement Learning with MATLAB 28 Nov • 7 min read. One of the restrictions is that the environment needs to be duplicated because we need to sample different trajectories starting from the same state. where www and sts_tst are 4×14 \times 14×1 column vectors. In terms of number of interactions, they are equally bad. Note that if we hit the 500 as episode length, we bootstrap on the learned value function. This inapplicabilitymay result from problems with uncertain state information. \end{aligned}∇w[21(Gt−V^(st,w))2]=−(Gt−V^(st,w))∇wV^(st,w)=−δ∇wV^(st,w). We optimize hyperparameters for the different approaches by running a grid search over the learning rate and approach-specific hyperparameters. past few years amazing results like learning to play Atari Games from raw pixels and Mastering the Game of Go have gotten a lot of attention This means that most of the parameters of the network are shared. As in my previous posts, I will test the algorithm on the discrete-cart pole environment. E[t=0∑T∇θlogπθ(at∣st)b(st)]=(T+1)E[∇θlogπθ(a0∣s0)b(s0)], I apologize in advance to all the researchers I may have disrespected with any blatantly wrong math up to this point. Kool, W., Van Hoof, H., & Welling, M. (2019). 13.5a One-Step Actor-Critic. REINFORCE with baseline. By this, we prevent to punish the network for the last steps although it succeeded. We compare the performance against: The number of iterations needed to learn is a standard measure to evaluate. It can be anything, even a constant, as long as it has no dependence on the action. If we have no assumption about R, then we can use REINFORCE with baseline bas in [1]: r wE[Rj ˇ w] = 1 2 E[(R b)(A E[AjX])Xjˇ w] (2) Denote was the update to weight wand as the learning rate, then the learning rule based on REINFORCE is given by: w =0 = (R b)(A E[AjX])X (3) 2. This can be improved by subtracting a baseline value from the Q values. Eighty-three male and female patients aged from 13 to 73 years were randomized to either of the following two treatment groups in a 1:1 ratio: satralizumab (120 mg) or placebo added to baseline … Reinforcement Learning (RL) refers to both the learning problem and the sub-field of machine learning which has lately been in the news for great reasons. A simple baseline, that looks similar to a trick commonly used in optimization literature, is to normalize the returns of each step of the episode by subtracting the mean and dividing by the standard deviation of returns at all time steps within the episode. \nabla_\theta J\left(\pi_\theta\right) &= \mathbb{E} \left[\sum_{t = 0}^T \nabla_\theta \log \pi_\theta \left(a_t \vert s_t \right) \sum_{t' = t}^T \left(\gamma^{t'} r_{t'} - b\left(s_t\right)\right) \right] \\ This will allow us to update the policy during the episode as opposed to after which should allow for faster training. LMMâââNeural Network That Animates Video Game Characters, Building an artificially intelligent system to augment financial analysis, Neural Networks from Scratch with Python Code and Math in Detailâ I, A Short Story of Faster R-CNNâs Object detection, Hello World-Implementing Neural Networks With NumPy, number of update steps (1 iteration = 1 episode + gradient update step), number interactions (1 interaction = 1 action taken in the environment), The regular REINFORCE loss, with the learned value as a baseline, The mean squared error between the learned value and the observed discounted return. &= -\delta \nabla_w \hat{V} \left(s_t,w\right) This output is used as the baseline and represents the learned value. A not yet explored benefit of sampled baseline might be for partially observable environments. Then we will show results for all different baselines on the deterministic environment. We do not use V in G. G is only the reward to go for every step in … Achetez et téléchargez ebook Reinforced Carbon Carbon (RCC) oxidation resistant material samples - Baseline coated, and baseline coated with tetraethyl orthosilicate (TEOS) impregnation (English Edition): Boutique Kindle - Science : Amazon.fr Starting from the state, we could also make the agent greedy, by making it take only actions with maximum probability, and then use the resulting return as the baseline. For example, for the LunarLander environment, a single run for the sampled baseline takes over 1 hour. &= \sum_s \mu\left(s\right) b\left(s\right) \nabla_\theta \sum_a \pi_\theta \left(a \vert s \right) \\ If we learn a value function that (approximately) maps a state to its value, it can be used as a baseline. However, the stochastic policy may take different actions at the same state in different episodes. However, the difference between the performance of the sampled self-critic baseline and the learned value function is small. The results on the CartPole environment are shown in the following figure. In the REINFORCE algorithm, Monte Carlo plays out the whole trajectory in an episode that is used to update the policy afterward. In contrast, the sample baseline takes the hidden parts of the state into account, as it will start from s=(a1,b). This technique, called whitening is often necessary for good optimization, especially in the deep learning setting. We have seen that using a baseline greatly increases the stability and speed of policy learning with REINFORCE. With enough motivation, let us now take a look at the Reinforcement Learning problem. Our main new result is to show that the gradient can be written in a form suitable for estimation from experience aided by an approximate action-value or advantage function. We see that the sampled baseline no longer gives the best results. We again plot the average episode length over 32 seeds, compared to the number of iterations as well as the number of interactions. Because Gt is a sample of the true value function for the current policy, this is a reasonable target. This can be even achieved with a single sampled rollout. Also, while most comparative studies focus on deterministic environments, we go one step further and analyze the relative strengths of the methods as we add stochasticity to our environment. It was soon discovered that subtracting a âbaselineâ from the return led to reduction in variance and allowed faster learning. This is considerably higher than for the previous two methods, suggesting that the sampled baseline give a much lower variance for the CartPole environment. As before, we also plotted the 25th and 75th percentile. Thus,those systems need to be modeled as partially observableMarkov decision problems which o… Instead, the model with the learned baseline performs best. Then we can train the states from our main trajectory based on the beam as baseline, but at the same time, use the states of the beam as well as training points, where the main trajectory serves as baseline. REINFORCE with Baseline Algorithm Initialize the actor μ (S) with random parameter values θμ. Also, the optimal policy is not unlearned in later iterations, which does regularly happen when using the learned value estimate as baseline. However, it does not solve the game (reach an episode of length 500). But assuming no mistakes, we will continue. The various baseline algorithms attempt to stabilise learning by subtracting the average expected return from the action-values, which leads to stable action-values. Download source code. Technically, any baseline would be appropriate as long as it does not depend on the actions taken. We saw that while the agent did learn, the high variance in the rewards inhibited the learning. One slight difference here is versus my previous implementation is that I’m implementing REINFORCE with a baseline value and using the mean of the returns as my baseline. &= \sum_s \mu\left(s\right) \sum_a \pi_\theta \left(a \vert s\right) \frac{\nabla_\theta \pi_\theta \left(a \vert s \right)}{\pi_\theta \left(a \vert s\right)} b\left(s\right) \\ The episode ends when the pendulum falls over or when 500 time steps have passed. \mathbb{E} \left[\nabla_\theta \log \pi_\theta \left(a_0 \vert s_0 \right) b\left(s_0\right) \right] &= \sum_s \mu\left(s\right) \sum_a \pi_\theta \left(a \vert s\right) \nabla_\theta \log \pi_\theta \left(a \vert s \right) b\left(s\right) \\ www is the weights parametrizing V^\hat{V}V^. The results for our best models from above on this environment are shown below. Wouter Kool University of Amsterdam ORTEC w.w.m.kool@uva.nl Herke van Hoof University of Amsterdam h.c.vanhoof@uva.nl Max Welling University of Amsterdam CIFAR m.welling@uva.nl ABSTRACT REINFORCE can be used to train models in structured prediction settings to di-rectly optimize the test-time objective. Now, we will implement this to help make things more concrete. It turns out that the answer is no, and below is the proof. The REINFORCE algorithm with baseline is mostly the same as the one used in my last post with the addition of the value function estimation and baseline subtraction. Applying this concept to CartPole, we have the following hyperparameters to tune: number of beams for estimating the state value (1, 2, and 4), the log basis of the sample interval (2, 3, and 4), and the learning rate (1e-4, 4e-4, 1e-3, 2e-3, 4e-3). Comparing all baseline methods together we see a strong preference for REINFORCE with the sampled baseline as it already learns the optimal policy before 200 iterations. Why does Java have support for time zone offsets with seconds precision? The outline of the blog is as follows: we first describe the environment and the shared model architecture. This would require 500*N samples which is extremely inefficient. The results were slightly worse than for the sampled one which suggests that exploration is crucial in this environment. For example, assume we have a two dimensional state space where only the second dimension can be observed. The optimal learning rate found by gridsearch over 5 different rates is 1e-4. Latest commit b2d179a Jun 11, 2019 History. REINFORCE with Baseline. Once we have sample a trajectory, we will know the true returns of each state, so we can calculate the error between the true return and the estimated value function as, δ=Gt−V^(st,w)\delta = G_t - \hat{V} \left(s_t,w\right) There has never been a better time for enterprises to harness its power, nor has the … Interestingly, by sampling multiple rollouts, we could also update the parameters on the basis of the jâth rollout. This method, which we call the self-critic with sampled rollout, was described in Kool et al.Â³ The greedy rollout is actually just a special case of the sampled rollout if you consider only one sample being taken by always choosing the greedy action. Stochasticity seems to make the sampled beams too noisy to serve as a good baseline. W. Zaremba et al., "Reinforcement Learning Neural Turing Machines", arXiv, 2016. this baseline is chosen as expected future reward given previous states/actions. However, the policy gradient estimate requires every time step of the trajectory to be calculated, while the value function gradient estimate requires only one time step to be calculated. Here, Gt is the discounted cumulative reward at time step t. Writing the gradient as an expectation over the policy/trajectory allows us to update the parameter similar to stochastic gradient ascent: As with any Monte Carlo based approach, the gradients of the REINFORCE algorithm suffer from high variance as the returns exhibit high variability between episodes - some episodes can end well with high returns whereas some could be very bad with low returns. I included the 12\frac{1}{2}21 just to keep the math clean. Reinforcement learning is probably the most general framework inwhich reward-related learning problems of animals, humans or machinecan be phrased. That is, it is not used for bootstrapping (updating the value estimate for a state from the estimated values of subsequent states), but only as a baseline for the state whose … 13.4 REINFORCE with Baseline. In our case this usually means that in more than 75% of the cases, the episode length was optimal (500) but that there were a small set of cases where the episode length was sub-optimal. Mark Saad in Reinforcement Learning with MATLAB 29 Nov • 6 min read. All together, this suggests that for a (mostly) deterministic environment, a sampled baseline reduces the variance of REINFORCE the best. We see that the learned baseline reduces the variance by a great deal, and the optimal policy is learned much faster. However, the method suffers from high variance in the gradients, which results in slow unstable learning and a lot of frustrationâ¦. In my next post, we will discuss how to update the policy without having to sample an entire trajectory first. Sign in with GitHub … Nevertheless, this improvement comes with the cost of increased number of interactions with the environment. Reinforcement Learning is the mos… By executing a full trajectory, you would know its true reward. Code: REINFORCE with Baseline. \nabla_\theta J\left(\pi_\theta\right) &= \mathbb{E} \left[\sum_{t = 0}^T \nabla_\theta \log \pi_\theta \left(a_t \vert s_t \right) \sum_{t' = t}^T \left(\gamma^{t'} r_{t'} - b\left(s_t\right)\right) \right] \\ In terms of number of iterations, the sampled baseline is only slightly better than regular REINFORCE. … We output log probabilities of the actions by using the LogSoftmax as the final activation function. However, more sophisticated baselines are possible. But this is just speculation and with some trial and error, a lower learning rate for the value function parameters might be more effective. Enjoy Afterpay, International Shipping and free delivery on orders over $100. This effect is due to the stochasticity of the policy. As a result, I have multiple gradient estimates of the value function which I average together before updating the value function parameters. And if none of the rollouts reach the goal, this means that all returns will be the same, and thus the gradient will be zero. Furthermore, in the environment with added stochasticity, we observed that the learned value function clearly outperformed the sampled baseline. spaces import Discrete, Box: def get_traj (agent, env, max_episode_steps, render, deterministic_acts = False): ''' Runs agent-environment loop for one whole episdoe (trajectory). layers as layers: from tqdm import trange: from gym. The REINFORCE with Baseline algorithm becomes. contrib. So I am not sure if the above results are accurate, or if there is some subtle mistake that I made. The research community is seeing many more promising results. &= \sum_s \mu\left(s\right) b\left(s\right) \sum_a \nabla_\theta \pi_\theta \left(a \vert s \right) \\ The figure shows that in terms of the number of interactions, sampling one rollout is the most efficient in reaching the optimal policy. Of course, there is always room for improvement. Besides, the log basis did not seem to have a strong impact, but the most stable results were achieved with log 2. But most importantly, this baseline results in lower variance, hence better learning of the optimal policy. The other methods suffer less from this issue because their gradients are mostly non-zero, and hence, this noise gives a better exploration for finding the goal. $89.95. In this way, if the obtained return is much better than the expected return, the gradients are stronger and vice-versa. Likewise, we substract a lower baseline for states with lower returns. V^(st,w)=wTst. The network takes the state representation as input and has 3 hidden layers, all of them with a size of 128 neurons. The capability of training machines to play games better than the best human players is indeed a landmark achievement. On the other hand, the learned baseline has not converged when the policy reaches the optimum because the value estimate is still behind. In the case of learned value functions, the state estimate for s=(a1,b) is the same as for s=(a2,b), and hence learns an average over the hidden dimensions. While most papers use these baselines in specific settings, we are interested in comparing their performance on the same task. The state is described by a vector of size 4, containing the position and velocity of the cart as well as the angle and velocity of the pole. w = w +\delta \nabla_w \hat{V} \left(s_t,w\right) This method more efficiently uses the information obtained from the interactions with the environmentâ´. We saw that while the agent did learn, the high variance in the rewards inhibited the learning. ∇θJ(πθ)=E[t=0∑T∇θlogπθ(at∣st)t′=t∑Tγt′rt′], Suppose we subtract some value, bbb, from the return that is a function of the current state, sts_tst, so that we now have, ∇θJ(πθ)=E[∑t=0T∇θlogπθ(at∣st)∑t′=tT(γt′rt′−b(st))]=E[∑t=0T∇θlogπθ(at∣st)∑t′=tTγt′rt′−∑t=0T∇θlogπθ(at∣st)b(st)]=E[∑t=0T∇θlogπθ(at∣st)∑t′=tTγt′rt′]−E[∑t=0T∇θlogπθ(at∣st)b(st)]\begin{aligned} But we also need a way to approximate V^\hat{V}V^. By contrast, Pigeon DRO8 showed clear evidence of symmetry: Its comparison-response rates were considerably higher on probe trials that reversed the symbolic baseline relations on which comparison responding was reinforced (positive trials) than on probe trials that reversed the symbolic baseline relations on which not-responding was reinforced (negative trials), F (1, 62) = … more info Size SIZE GUIDE. Policy gradient is an approach to solve reinforcement learning problems. In my last post, I implemented REINFORCE which is a simple policy gradient algorithm. We can explain this by the fact that the learned value function can learn to give an expected/averaged value in certain states. w=w+(Gt−wTst)st. REINFORCE with Baseline Policy Gradient Algorithm. For each training episode, generate the episode experience by following the actor policy μ (S). BUY 4 REINFORCE SAMPLES, GET A BASELINE FOR FREE! The division by stepCt could be absorbed into the learning rate. We want to minimize this error, so we update the parameters using gradient descent: w=w+δ∇wV^(st,w)\begin{aligned} In a stochastic environment, the sampled baseline would thus be more noisy. REINFORCE with sampled baseline: the average return over a few samples is taken to serve as the baseline. With advancements in deep learning, these algorithms proved very successful using powerful networks as function approximators. This enables the gradients to be non-zero, and hence can push the policy out of the optimum which we can see in the plot above. The source code for all our experiments can be found here: Rennie, S. J., Marcheret, E., Mroueh, Y., Ross, J., & Goel, V. (2017). If you haven’t looked into the field of reinforcement learning, please first read the section “A (Long) Peek into Reinforcement Learning » Key Concepts”for the problem definition and key concepts. The goal is to keep the pendulum upright by applying a force of -1 or +1 (left or right) to the cart. Implementation of REINFORCE with Baseline algorithm, recreation of figure 13.4 and demonstration on Corridor with switched actions environment. Note that whereas this is a very common technique, the gradient is no longer unbiased. Amongst all the approaches in reinforcement learning, policy gradient methods received a lot of attention as it is often easier to directly learn the policy without the overhead of learning value functions and then deriving a policy.

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